Approximation of penalty terms in Tikhonov functionals-theory and applications in inverse problems

摘要

One feasible way to minimize a non-smooth functional is to replace it by some smoothed version, which leads to a surrogate minimization problem that is easily treated by standard means. A prominent example of such a problem is given by a Tikhonov-type functional incorporating a sparsity-enforcing penalty term. It has received enormous attention in recent years, yet its efficient minimization remains challenging. In this paper we consider general Tikhonov-type functionals and show, under mild conditions, the stability of their minimizer with respect to the replacement of the penalty term with an appropriate approximation. In particular, we consider the case of separable penalty terms. Finally, we apply the proposed strategy to the inverse medium problem and demonstrate numerical results that indicate the efficiency of the approach.